Journal of the ICRU Vol 7 No 1 (2007) Report 77 Oxford University Press
doi:10.1093/jicru/ndm002
ELASTIC SCATTERING OF ELECTRONS AND POSITRONS
CONTENTS 1
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
EXECUTIVE SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
LIST OF ABBREVIATIONS AND ACRONYMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.1 Applications of data to be given . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Quantities and units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 General characteristics of cross-sections for elastic scattering of electrons and positrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Broader perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Scope of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Materials treated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Range of kinetic energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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13 13 14 14 15
2 EXPERIMENTAL METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.1 Gases (atoms and molecules) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Electron-swarm experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Attenuation and crossed-beam experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Comparison of the two techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Condensed phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Electron-energy regions of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Electron energies lower than 30 eV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.1 Low-energy electron transmission spectroscopy (LEET). . . . . . . . . . 2.2.2.2 High-resolution reflectivity measurements . . . . . . . . . . . . . . . . . . . . 2.2.2.3 Method for analysis of high-resolution reflectivity measurements . . 2.2.3 Electron energies between 100 eV and 100 keV. . . . . . . . . . . . . . . . . . . . . . . . 2.2.3.1 Measurement of electron inelastic mean-free paths by elastic-peak electron spectroscopy (EPES) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3.2 Analysis of elastic-scattering data from single-crystal surfaces and adsorbates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3.3 Measurement of electron attenuation lengths by Auger-electron spectroscopy and x-ray photoelectron spectroscopy . . . . . . . . . . . . . . 2.2.3.4 Measurement of electron backscattering yields and the energy distributions and angular distributions of backscattered electrons .
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PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ELASTIC SCATTERING OF ELECTRONS AND POSITRONS
2.2.3.5 Measurement of the angular distribution of photoelectrons. . . . . . . . . . . 2.2.3.6 Measurement of x-ray spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3.7 Measurement of Auger-electron spectra, x-ray photoelectron spectra, and reflection electron energy-loss spectra . . . . . . . . . . . . . . . . . . . . . . . .
29 30
3 THEORETICAL BACKGROUND. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 CALCULATIONS FOR ATOMS USED FOR THE DATA GENERATION . . . . . . . . . . . . . . . . . . . . . .
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4.1 Choice of the interaction potential for atoms. . . 4.2 Numerical calculation methods. . . . . . . . . . . . . . . 4.2.1 Solution of the radial equations. . . . . . . . . . . . 4.2.2 The reduced-series method. . . . . . . . . . . . . . . . 4.3 Properties of the phase shifts . . . . . . . . . . . . . . . . 4.4 High-energy factorization . . . . . . . . . . . . . . . . . . . 4.4.1 High-energy screening . . . . . . . . . . . . . . . . . . . 4.4.2 Finite nuclear-size effects . . . . . . . . . . . . . . . . . 4.5 Cross-sections of atoms. . . . . . . . . . . . . . . . . . . . . . 4.6 Comparison with more elaborate calculations . .
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5 EXPERIMENTAL DATA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... 5 keV ..... ..... ..... ..... ..... ..... ..... .....
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5.1 Atoms and molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Differential cross sections of atoms for electron energies 5.1.1.1 Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1.2 Noble gases and mercury . . . . . . . . . . . . . . . . . 5.1.2 Total cross-sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Spin polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Electron energies higher than 50 keV. . . . . . . . . . . . . . . 5.1.5 Electron scattering by water molecules . . . . . . . . . . . . . 5.2 Results from condensed-phase experiments . . . . . . . . . . . 5.2.1 Mean-free paths determined from LEET experiments . .
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3.1 Basic concepts and formal theory of scattering . . . . . . . . . . . . . 3.2 The effective interaction potential for atoms . . . . . . . . . . . . . . . 3.2.1 Nuclear charge distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Screening of the nuclear charge by atomic electrons . . . . . . . . 3.2.3 Electron-exchange effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Positron-specific features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4.1 The sign of the interaction potential . . . . . . . . . . . . . 3.2.4.2 Positronium formation . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4.3 Positron annihilation . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Differential cross-section and spin polarization. . . . . . . . . . . . 3.3 Optical-model potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Correlation-polarization potential . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Absorption potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Elastic scattering by ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Elastic scattering by molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Independent-atom approximation . . . . . . . . . . . . . . . . . . . . . . 3.6 Scattering in the condensed phase . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Elastic scattering from single crystals . . . . . . . . . . . . . . . . . . . 3.6.2 Elastic scattering from disordered solids. . . . . . . . . . . . . . . . . 3.6.3 Coherent low-energy electron scattering in the solid phase . . . 3.6.4 Incoherent low-energy electron scattering in the solid phase . . 3.7 Resonance electron scattering in the condensed phase . . . . . .
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CONTENTS
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6 MULTIPLE-SCATTERING ANGULAR DEFLECTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6.1 Goudsmit – Saunderson multiple-scattering distribution . . . . . . . . . . . . . . 6.1.1 The Wentzel DCS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Multiple scattering distributions with energy loss . . . . . . . . . . . . . . . . . . . 6.3 Contribution of inelastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Influence of elastic-scattering data on multiple-scattering calculations.
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7 DATA PRESENTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7.1 Validation of the code system ELSEPA . . 7.2 Energy range of validity . . . . . . . . . . . . 7.3 Structure of the database . . . . . . . . . . . 7.3.1 Auxiliary programs . . . . . . . . . . . . 7.4 Graphical interface . . . . . . . . . . . . . . . .
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8 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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APPENDIX A RELATIVISTIC KINEMATICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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APPENDIX B THE DIRAC EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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B.1 Dirac plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.1 Polarized electron beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Central fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
126 128 129
APPENDIX C QUANTUM THEORY OF SCATTERING BY A CENTRAL POTENTIAL . . . . . . . . . . . .
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C.1 Non-relativistic (Schro¨dinger) theory . . . . . C.1.1 Partial-wave series . . . . . . . . . . . . . . . . . C.1.2 Born approximation . . . . . . . . . . . . . . . . C.1.3 WKB approximation for the phase shifts. C.1.4 The eikonal approximation . . . . . . . . . . . C.1.5 Scattering by a Coulomb field . . . . . . . . . C.2 Relativistic (Dirac) theory . . . . . . . . . . . . . . . C.2.1 Partial-wave expansion . . . . . . . . . . . . . . C.2.2 Differential cross-section . . . . . . . . . . . . . C.2.3 Born approximation . . . . . . . . . . . . . . . . C.2.4 WKB approximation for the phase shifts.
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5.2.2 Scattering probabilities per unit length and cross-sections determined from HREEL experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Relationship between quasi-elastic scattering and the electronic band-structure of solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Indirect validation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Determinations of electron inelastic mean-free paths by elastic-peak electron spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Analysis of low-energy electron diffraction measurements . . . . . . . . . . . . . . . 5.3.3 Measurement of backscattered yields and the energy and angular distributions of backscattered electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Measurements of angular distributions of photoelectrons . . . . . . . . . . . . . . . 5.3.5 Measurements of x-ray spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Measurements of Auger-electron spectra and x-ray photoelectron spectra . . . 5.3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ELASTIC SCATTERING OF ELECTRONS AND POSITRONS
C.2.5 The eikonal approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2.6 Scattering by a Coulomb field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2.7 Modified Coulomb fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
144 145 148
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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doi:10.1093/jicru/ndm002
ELASTIC SCATTERING OF ELECTRONS AND POSITRONS
CONTENTS 1
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
EXECUTIVE SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
LIST OF ABBREVIATIONS AND ACRONYMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.1 Applications of data to be given . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Quantities and units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 General characteristics of cross-sections for elastic scattering of electrons and positrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Broader perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Scope of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Materials treated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Range of kinetic energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 EXPERIMENTAL METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.1 Gases (atoms and molecules) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Electron-swarm experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Attenuation and crossed-beam experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Comparison of the two techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Condensed phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Electron-energy regions of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Electron energies lower than 30 eV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.1 Low-energy electron transmission spectroscopy (LEET). . . . . . . . . . 2.2.2.2 High-resolution reflectivity measurements . . . . . . . . . . . . . . . . . . . . 2.2.2.3 Method for analysis of high-resolution reflectivity measurements . . 2.2.3 Electron energies between 100 eV and 100 keV. . . . . . . . . . . . . . . . . . . . . . . . 2.2.3.1 Measurement of electron inelastic mean-free paths by elastic-peak electron spectroscopy (EPES) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3.2 Analysis of elastic-scattering data from single-crystal surfaces and adsorbates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3.3 Measurement of electron attenuation lengths by Auger-electron spectroscopy and x-ray photoelectron spectroscopy . . . . . . . . . . . . . . 2.2.3.4 Measurement of electron backscattering yields and the energy distributions and angular distributions of backscattered electrons .
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PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ELASTIC SCATTERING OF ELECTRONS AND POSITRONS
2.2.3.5 Measurement of the angular distribution of photoelectrons. . . . . . . . . . . 2.2.3.6 Measurement of x-ray spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3.7 Measurement of Auger-electron spectra, x-ray photoelectron spectra, and reflection electron energy-loss spectra . . . . . . . . . . . . . . . . . . . . . . . .
29 30
3 THEORETICAL BACKGROUND. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
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32 33 34 35 39 40 40 41 41 41 42 43 44 45 45 46 47 47 50 52 52 53
4 CALCULATIONS FOR ATOMS USED FOR THE DATA GENERATION . . . . . . . . . . . . . . . . . . . . . .
57
4.1 Choice of the interaction potential for atoms. . . 4.2 Numerical calculation methods. . . . . . . . . . . . . . . 4.2.1 Solution of the radial equations. . . . . . . . . . . . 4.2.2 The reduced-series method. . . . . . . . . . . . . . . . 4.3 Properties of the phase shifts . . . . . . . . . . . . . . . . 4.4 High-energy factorization . . . . . . . . . . . . . . . . . . . 4.4.1 High-energy screening . . . . . . . . . . . . . . . . . . . 4.4.2 Finite nuclear-size effects . . . . . . . . . . . . . . . . . 4.5 Cross-sections of atoms. . . . . . . . . . . . . . . . . . . . . . 4.6 Comparison with more elaborate calculations . .
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5 EXPERIMENTAL DATA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... 5 keV ..... ..... ..... ..... ..... ..... ..... .....
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57 58 58 60 62 65 65 69 74 75
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5.1 Atoms and molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Differential cross sections of atoms for electron energies 5.1.1.1 Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1.2 Noble gases and mercury . . . . . . . . . . . . . . . . . 5.1.2 Total cross-sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Spin polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Electron energies higher than 50 keV. . . . . . . . . . . . . . . 5.1.5 Electron scattering by water molecules . . . . . . . . . . . . . 5.2 Results from condensed-phase experiments . . . . . . . . . . . 5.2.1 Mean-free paths determined from LEET experiments . .
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77 77 77 77 78 79 81 83 87 88
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3.1 Basic concepts and formal theory of scattering . . . . . . . . . . . . . 3.2 The effective interaction potential for atoms . . . . . . . . . . . . . . . 3.2.1 Nuclear charge distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Screening of the nuclear charge by atomic electrons . . . . . . . . 3.2.3 Electron-exchange effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Positron-specific features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4.1 The sign of the interaction potential . . . . . . . . . . . . . 3.2.4.2 Positronium formation . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4.3 Positron annihilation . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Differential cross-section and spin polarization. . . . . . . . . . . . 3.3 Optical-model potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Correlation-polarization potential . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Absorption potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Elastic scattering by ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Elastic scattering by molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Independent-atom approximation . . . . . . . . . . . . . . . . . . . . . . 3.6 Scattering in the condensed phase . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Elastic scattering from single crystals . . . . . . . . . . . . . . . . . . . 3.6.2 Elastic scattering from disordered solids. . . . . . . . . . . . . . . . . 3.6.3 Coherent low-energy electron scattering in the solid phase . . . 3.6.4 Incoherent low-energy electron scattering in the solid phase . . 3.7 Resonance electron scattering in the condensed phase . . . . . .
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CONTENTS
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6 MULTIPLE-SCATTERING ANGULAR DEFLECTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
6.1 Goudsmit – Saunderson multiple-scattering distribution . . . . . . . . . . . . . . 6.1.1 The Wentzel DCS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Multiple scattering distributions with energy loss . . . . . . . . . . . . . . . . . . . 6.3 Contribution of inelastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Influence of elastic-scattering data on multiple-scattering calculations.
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7 DATA PRESENTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7.1 Validation of the code system ELSEPA . . 7.2 Energy range of validity . . . . . . . . . . . . 7.3 Structure of the database . . . . . . . . . . . 7.3.1 Auxiliary programs . . . . . . . . . . . . 7.4 Graphical interface . . . . . . . . . . . . . . . .
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8 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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APPENDIX A RELATIVISTIC KINEMATICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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APPENDIX B THE DIRAC EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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B.1 Dirac plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.1 Polarized electron beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Central fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
126 128 129
APPENDIX C QUANTUM THEORY OF SCATTERING BY A CENTRAL POTENTIAL . . . . . . . . . . . .
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C.1 Non-relativistic (Schro¨dinger) theory . . . . . C.1.1 Partial-wave series . . . . . . . . . . . . . . . . . C.1.2 Born approximation . . . . . . . . . . . . . . . . C.1.3 WKB approximation for the phase shifts. C.1.4 The eikonal approximation . . . . . . . . . . . C.1.5 Scattering by a Coulomb field . . . . . . . . . C.2 Relativistic (Dirac) theory . . . . . . . . . . . . . . . C.2.1 Partial-wave expansion . . . . . . . . . . . . . . C.2.2 Differential cross-section . . . . . . . . . . . . . C.2.3 Born approximation . . . . . . . . . . . . . . . . C.2.4 WKB approximation for the phase shifts.
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131 132 134 135 136 137 138 139 140 141 143
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5.2.2 Scattering probabilities per unit length and cross-sections determined from HREEL experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Relationship between quasi-elastic scattering and the electronic band-structure of solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Indirect validation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Determinations of electron inelastic mean-free paths by elastic-peak electron spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Analysis of low-energy electron diffraction measurements . . . . . . . . . . . . . . . 5.3.3 Measurement of backscattered yields and the energy and angular distributions of backscattered electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Measurements of angular distributions of photoelectrons . . . . . . . . . . . . . . . 5.3.5 Measurements of x-ray spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Measurements of Auger-electron spectra and x-ray photoelectron spectra . . . 5.3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ELASTIC SCATTERING OF ELECTRONS AND POSITRONS
C.2.5 The eikonal approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2.6 Scattering by a Coulomb field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2.7 Modified Coulomb fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
144 145 148
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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